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 A056193 Goodstein sequence with a(2)=4: to calculate a(n+1), write a(n) in the hereditary representation base n, then bump the base to n+1, then subtract 1. 6

%I

%S 4,26,41,60,83,109,139,173,211,253,299,348,401,458,519,584,653,726,

%T 803,884,969,1058,1151,1222,1295,1370,1447,1526,1607,1690,1775,1862,

%U 1951,2042,2135,2230,2327,2426,2527,2630,2735,2842,2951,3062,3175,3290,3407

%N Goodstein sequence with a(2)=4: to calculate a(n+1), write a(n) in the hereditary representation base n, then bump the base to n+1, then subtract 1.

%C Goodstein's theorem shows that such a sequence is finite (i.e. eventually reaches 0) for any starting value [e.g. if a(2)=1 then a(3)=0; if a(2)=2 then a(5)=0; and if a(2)=3 then a(7)=0]. With a(2)=4 we have a(3*2^(3*2^27+27)-1)=0, which is well beyond the 10^(10^8)-th term.

%C The second half of such sequences is declining and the previous quarter is stable.

%C The resulting sequence 2,3,5,7,3*2^402653211 - 1, ... (see Comments in A056041) grows too rapidly to have its own entry.

%D Goodstein, R. L., On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.

%H _Reinhard Zumkeller_, <a href="/A056193/b056193.txt">Table of n, a(n) for n = 2..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoodsteinSequence.html">Goodstein Sequence.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>

%H _Reinhard Zumkeller_, <a href="/A211378/a211378.hs.txt">Haskell programs for Goodstein sequences</a>

%e a(2)=4=2^2, a(3)=3^3-1=26=2*3^2+2*3+2, a(4)=2*4^2+2*4+2-1=41=2*4^2+2*4+1, a(5)=2*5^2+2*5+1-1=60=2*5^2+2*5, a(6)=2*6^2+2*6-1=83=2*6^2+6+5, a(7)=2*7^2+7+5-1=109 etc.

%Y Cf. A056041, A056004, A059934, A057650, A059933, A059935, A059936.

%Y Cf. A215409, A222117, A211378.

%K fini,nonn

%O 2,1

%A _Henry Bottomley_, Aug 02 2000

%E Edited by _N. J. A. Sloane_, Mar 06 2006

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